I tested new BlizzardUI_Arena a little and want to share my findings.
First of all - Pre-start specialization detection work only on normal arenas. Wargames don't display it.

Every time when specialization data of somebody in enemy team will become available ARENA_PREP_OPPONENT_SPECIALIZATIONS event is triggered.

After that it is possible to get specialization id by using GetArenaOpponentSpec(x).
Where x is 1...5. The numbering corresponds Arena units so GetArenaOpponentSpec(3) will get specialization id of Arena3. Check UpdatePrepFrames() function in Blizzard_ArenaUI.lua for more details.

When we have specialization id we can use GetSpecializationInfoByID(number) and recive:

Specialization id (Yes, again)

Localized name

Localized long description used in Talent UI

Path to specialization icon

Name of background used in Talent UI

Specialization role

Non-localized class token

Sadly there is no new way to get specializations of your own team.Inpecting entire party is still required.GetInspectSpecialization(inspectedUnit) will return specialization id.